Understanding of quantum systems, whose interaction to their environment is non negligible, have raised great interest recently, mostly due to their central role in the nascent field of quantum computation. Classical numerical simulation of open quantum systems is a difficult problem due to the exponentially large Liouville space of density operators. Tensor network state (TNS) methods, regularly used nowadays to study closed systems, provide a promising tool to study open quantum systems too. The PhD student will join this research as a member of the Strongly Correlated Systems ‘Lendület’ Research Group. The main objective of the PhD research is to develop TNS algorithms that are capable to target stationary and dynamical quantum states of open quantum systems with short- and long-ranged interactions in one- and higher dimensions. The developed methods can then be used to study strongly correlated quantum models, like condensed matter, chemical, and nuclear systems. The proposed research requires both strong theoretical and numerical skills and interest in the field of quantum many-body systems.